Problem 65 - Prealgebra

Question

A ladder is leaning against the top of a 15 foot wall. If the bottom of the ladder is 20 feet from the wall, how long is the ladder?

Step-by-Step Solution

Answer

The length of the ladder is 25 feet.

Step 1: Identify the right triangle

The ladder, the wall, and the ground form a right triangle. The wall is the vertical side, the distance from the bottom of the wall to the ladder is the horizontal side, and the ladder itself is the hypotenuse.

Step 2: Apply the Pythagorean Theorem

To find the length of the ladder, use the Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse.

Step 3: Substitute the known values into the formula

Substitute 15 feet for \(a\) (the height of the wall) and 20 feet for \(b\) (the distance from the wall to the base of the ladder) into \(a^2 + b^2 = c^2\). So, \(15^2 + 20^2 = c^2\).

Step 4: Calculate \(a^2\) and \(b^2\)

Compute \(15^2 = 225\) and \(20^2 = 400\). This gives: \(225 + 400 = c^2\).

Step 5: Solve for \(c^2\)

Add the results from Step 4: \(225 + 400 = 625\). Thus, \(c^2 = 625\).

Step 6: Find \(c\), the hypotenuse

Take the square root of both sides to solve for \(c\): \(c = \sqrt{625} = 25\). This means the length of the ladder is 25 feet.

Key Concepts

Understanding Right Triangles

In geometry, a right triangle is a triangle that has one of its angles equal to 90 degrees. This makes it unique compared to other triangles, because the two sides that form the 90-degree angle are called the legs, and the side opposite the 90-degree angle is called the hypotenuse. In the given exercise, the problem describes the situation involving a ladder leaning against a wall. Here, we can identify: - The wall serves as one leg because it is vertical. - The distance from the ground directly beneath the ladder to the bottom of the wall is the other leg as it is horizontal. - The ladder itself, which opposes the right angle formed by the wall and the ground, acts as the hypotenuse. Recognizing right triangles in real-life problems is crucial as it helps apply the Pythagorean theorem effectively. You'll often encounter right triangles in architecture, art, and everyday objects, making this knowledge very practical!

The Role of the Hypotenuse

The hypotenuse is the star of the right triangle! It's the longest side and directly opposite the right angle. In this scenario, the hypotenuse is the ladder that needs measuring.When using the Pythagorean theorem, denoted as \[a^2 + b^2 = c^2\], where:- \(a\) and \(b\) are the triangle's legs - \(c\) represents the hypotenuse We need to find \(c\), which means the length of the ladder. Knowing the hypotenuse's role is essential because it triggers the use of this specific geometric rule, which couldn't apply otherwise. It's vital whenever you’re solving any problem involving right triangles to identify and distinguish the hypotenuse for accurate calculations and interpretations.

Effective Problem-Solving with the Pythagorean Theorem

Solving word problems that involve right triangles can initially seem daunting, but it becomes straightforward with the right approach. Here’s how you can effectively tackle these problems using the Pythagorean theorem:1. **Understand the Problem**: Visualize the scenario. Identify the elements forming the right triangle, often labeled in textbooks as a "diagramming" stage.2. **Identify the Right Triangle Elements**: Recognize which sides are the legs and which side is the hypotenuse.3. **Use the Pythagorean Theorem**: With the formula \[a^2 + b^2 = c^2\], plug in the known values. In our ladder example: - Let \(a = 15\) (height of the wall) - Let \(b = 20\) (distance to the wall)4. **Calculate**: Perform the computations for the squares of \(a\) and \(b\). - Compute: \(15^2 = 225\) - Compute: \(20^2 = 400\) - Sum these results: \(225 + 400 = 625\)5. **Solve for the Hypotenuse**: Take the square root. - \(c = \sqrt{625} = 25\)Practicing this method does more than solve the problem; it builds a strong mathematical foundation, boosts confidence, and enhances logical reasoning skills.